Question

find the sum of first seventeen numbers of an ap whose 4th and 9th tearm are -15 and -30

Answer 1

Answer:

The sum of first seventeen terms is - 510

Step-by-step explanation:

Given as :

The 4th term of an A.P =  $t_4$ =  - 15

The 9th term of an A.P =  $t_9$ =  - 30

For an arithmetic progression

The nth term is given as $t_n$ = a + ( n - 1)×d

Where a is the first term and d is the common difference between numbers

So, For 4th term

$t_n$ = a + ( n - 1)×d

Or, $t_4$ = a + ( n - 1)×d

- 15 = a + ( 4 - 1)×d

Or, - 15 = a + 3 d            .........1

So, For 9th term

$t_n$ = a + ( n - 1)×d

$t_9$ = a + ( n - 1)×d

- 30 = a + ( 9 - 1)×d

Or, - 30 = a + 8 d            .........2

Solve eq 1 and 2

( a + 8 d ) - ( a + 3 d ) = - 30 - ( - 15)

or, ( a - a ) + ( 8 d - 3 d ) = - 30 + 15

or, 0 + 5 d = - 15

∴ d = - $\frac{15}{5}$ = - 3

Now, put the value of d in eq 1

I.e - 15 = a + 3 × ( - 3)  

Or. - 15 = a - 9

∴ a = -15 + 9 = - 6

Now The sum of nth term is written as :

$s_n$ = $\frac{n}{2}$ × [ 2 × a + ( n - 1 )×d ]

Where n is the nth term

a is the first term

d is the common difference

So For n = 17th term

$s_17$ = $\frac{17}{2}$ × [ 2 × ( - 6) + ( 17 - 1 )×( - 3) ]

Or, $s_17$ = $\frac{17}{2}$ × [ - 12 - 48 ]

Or, $s_17$ = $\frac{17}{2}$ × ( - 60 )

Or, $s_17$ = 17 × ( - 30)

∴  $s_17$ = - 510

Hence The sum of first seventeen terms is - 510  Answer